Question

The owner of a ranch has 2 , 100 2,100 yards of fencing material which to enclose a rectangular piece of grazing land along a straight portion of a river. If fencing is not required along the river, what are the dimensions of the pasture having the largest area

Answer 1

x = 1050 yd

y = 525 yd

A(max) = 551250 yd²

Explanation:

For enclosing a rectangular area (only three sides, since one side will be the river) we have 2100 yards, then the length of fencing material is:

L = 2100 = x + 2y ⇒ y = ( 2100 - x ) / 2

Where x and y are the sides of the rectangle ( x is the parallel side to the river)

The area of the rectangle is:

A = x*y

And as y = (2100 - x ) / 2

We can express A as a function of x, getting:

A(x) = x* (2100 - x ) /2 or

A(x) =( 2100*x - x² )/ 2 ⇒ A(x) = 1050*x - (1/2)*x² (1)

Taking derivatives on both sides of the equation we have

A´(x) = 1050 - x

A´(x) = 0 means 1050 - x = 0

x = 1050 yd

And as A´´(x) = - 1 A´´(x) < 0

We have a maximum for the function at the point x = 1050

Now

y = ( 2100 - x ) /2 then

y = ( 2100 - 1050 ) / 2

y = 525 yd

And

A(max) = 1050* 525

A(max) = 551250 yd²